Re: ** says: Definition: sum{i in N} i = 0

On 24 Jun., 18:08, hagman <goo...@xxxxxxxxxxxxx> wrote:
On 24 Jun., 10:32, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:





On 24 Jun., 02:33, Subluxian <cbr...@xxxxxxxxxxxxxxxxx> wrote:

But in order to satisfy your desire: What about my claim

sqrt(2) = 7 ?

Of course it is impossible to redefine the symbol sqrt(2), which is
undefined in N, by investigating 7^2 = 2.

But when you define "sqrt(2) := 7" in the context of the naturals, you
are /not/ also asserting that 7^2 = 2. The only reason that sqrt(2) is
available for an extension of the definition is precisely because
there is /no/ natural m such that m^2 = 2

We can accept your definition,

No. We cannot accept this definition - not inmathematics.

and then it is no longer a theorem that
"for all n in N, if sqrt(n) exists then sqrt(n)*sqrt(n) = n". Instead,
the theorem would be: "for all n in N with n not equal to 2, if
sqrt(n) exists then sqrt(n)*sqrt(n) = n".

But it was /never/ a theorem /before/ your extension to the definition
that "for all n in N, sqrt(n)*sqrt(n)=n". In particular, it was /
never/ a theorem that "sqrt(2)*sqrt(2)=2"; because sqrt(2) is not
defined in this context, and thus the product "sqrt(2)*sqrt(2)" is not
defined in this context.

But it was a theorem before and must remain a theorem that all square
roots s > 1 which are in N (like s = 7 is in N) are less than their
square. Here we would have 7^2 = 2 and as 2 < 7 this is wrong.

First, "their square" is a bad formulation in this context.

That's why I invented it!
Virgil keep quiet!

Defining sqrt(2)=7 does not imply 7^2=2.

What do you think, is the meaning of sqrt? Any idea??

Anyway order relations for squares and square roots are
very prone to losing theorem status in any extension.

You can extend what you like, but you may not confuse it with orthodox
mathematics.
Sum = r in R implies that there is a series the partial sums of which
converge according to Cauch .

For example x^2 >= x is a theorem in N, but not in Q as (1/2)^2 < 1/4.
And x^2 >=0, which is valid in Q and R does not hold for the imaginary
unit.



Similarly, it is not a theorem in analysis that "if f(n) > 0 for all
n, then sum(n e N) f(n) > 0". The theorem instead is "if f(n) > 0 for
all n, then if sum(n e N) f(n) exists /as the limit of a Cauchy
series/, then sum(n e N) f(n) > 0".

That is an equivalence. Therefore: If sum(n e N) f(n) exists, then it
is the limit of the Cauchy sequence of its partial sums.

When speaking of limits of partial sums, you I would prefer you to
write
sum_{n=1}^{\infty} f(n) instead of sum_{n e N} f(n).
The former looks like a generalisation of sum_{n=1}^m f(n), which is
the sum of all
f(n) starting from f(1) and up to including f(m); apparently the
symbol
that was chosen is not perfect since the limit it represents can
hardly
be described as "the sum of all f(n) starting from f(1) and up to
including f(\infty)"
However, the other notation suggests that it is a special case of
sum_{n e S} f(n)
where S is an arbitrary set and this would suit your purposes at most
for absolutely convergent series.
In all other cases of convergence, you need the linear order of N,
which
is implied in the other notation which gives a lower and an upper
index limit.



Thus, ***'s redefinition does not cause a contradiction with either
the former non-theorem, nor the latter theorem.

It is a theorem in analysis that every sequence which has a limit is
converging to this limit. (It is not only a theorem but is a fact of
axiom-independent mathematics.)

The subject was "sums" not "limits".



It is not the square root
which I give but an hitherto undefined sqrt(2)-symbol? Got it?

Sure; it's the same thing as ***'s redefinition,

Nice to hear that, thank you.

and causes no
problems - as long as you're explicit with your definitions, and
consistent with their application.

It causes problems, if we do not distinguish. If *** had stated: sum*
= 0 or even better XYZ = 0, nobody would have objected. But he
wilfully used the usual term for Cauchy series, namely sum{n in N},

The capital sigma is originally not related to convergence!

It was meant for converging series and such which do not converge, but
not for nonconverging series which have a real value. Nobody did
consider this because it was too absurd an idea, nearly as absurd as
the finished infinity.

Therefore one should already have come up with a new symbol
for the limit of convergent partial sums.
As a matter of fact, there *is* a notational difference between

sum_{n=1}^{\infty} a_n

as used for convergent series and

sum_{n e N} a_n

as (IIRC) used by ***.

There is a difference, in principle, because in the second case there
is no order implied. Nevertheless this difference vanished for
positive series because they converge absolutely or they diverge
absolutely.

without indicating that he did not mean converging series.

If it was stated as a definition and not as a theorem, it
could hardly have meant converging series.

It is an example of a wrong definition. If I define sqrt(2) = 7 or 2 +
2 = 5, then that's rubbish and nothing else.
Virgil, keep quiet!

Cantor said: "Hypothesen" welche gegen diese Grundwahrheiten
verstoßen, sind ebenso falsch und widersprechend, wie etwa der Satz 2
+ 2 = 5 oder ein viereckiger Kreis. Es genügt für mich, derartige
Hypothesen an die Spitze irgend einer Untersuchung gestellt zu sehen,
um von vorn herein zu wissen, daß diese Untersuchung falsch sein muss.
(Hypothesen here means axioms or defintions.)

Namely as wrong as, for example, 0 = sum_{n e N} n < sum_{n e N} 1 =
omega.

Regards, WM

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